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Mirrors > Home > MPE Home > Th. List > ltleii | Structured version Visualization version GIF version |
Description: 'Less than' implies 'less than or equal to' (inference). (Contributed by NM, 22-Aug-1999.) |
Ref | Expression |
---|---|
lt.1 | ⊢ 𝐴 ∈ ℝ |
lt.2 | ⊢ 𝐵 ∈ ℝ |
ltlei.1 | ⊢ 𝐴 < 𝐵 |
Ref | Expression |
---|---|
ltleii | ⊢ 𝐴 ≤ 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltlei.1 | . 2 ⊢ 𝐴 < 𝐵 | |
2 | lt.1 | . . 3 ⊢ 𝐴 ∈ ℝ | |
3 | lt.2 | . . 3 ⊢ 𝐵 ∈ ℝ | |
4 | 2, 3 | ltlei 10843 | . 2 ⊢ (𝐴 < 𝐵 → 𝐴 ≤ 𝐵) |
5 | 1, 4 | ax-mp 5 | 1 ⊢ 𝐴 ≤ 𝐵 |
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